Results 1  10
of
43
Isoperimetric and universal inequalities for eigenvalues, in Spectral Theory and Geometry
 London Mathematical Society Lecture Note Series
, 1999
"... PaynePólyaWeinberger conjecture, Sperner’s inequality, biharmonic operator, biLaplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the PólyaSzegő conjecture, universal inequalities for eigenvalues, HileProtter inequality, H. C. Yang’s inequality. Short title: Isoperimetric ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
PaynePólyaWeinberger conjecture, Sperner’s inequality, biharmonic operator, biLaplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the PólyaSzegő conjecture, universal inequalities for eigenvalues, HileProtter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and biLaplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to
Generalized AlonBoppana theorems and errorcorrecting codes
 Journal of Discrete Mathematics
, 2002
"... In this paper we describe several theorems that give lower bounds on the second eigenvalue of any quotient of a given size of a fixed graph, G. These theorems generalize AlonBoppana type theorems, where G is a regular (infinite) tree. When G is a hypercube, our theorems give minimum distance upper ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
In this paper we describe several theorems that give lower bounds on the second eigenvalue of any quotient of a given size of a fixed graph, G. These theorems generalize AlonBoppana type theorems, where G is a regular (infinite) tree. When G is a hypercube, our theorems give minimum distance upper bounds on linear binary codes of a given size and information rate. Our bounds at best equal the current best bounds for codes, and only apply to linear codes. However, it is of interest to note that (1) one very simple AlonBoppana argument yields nontrivial code bound, and (2) our AlonBoppana argument that equals a current best bound for codes has some hope of improvement. We also improve the bound in sharpest known AlonBoppana theorem (i.e., when G is a regular tree). 1
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Lecture Notes On Geometric Analysis
, 1996
"... Contents 0 Introduction 1 First and Second Variational Formulas for Area 2 Bishop Comparison Theorem 3 BochnerWeitzenbock Formulas 4 Laplacian Comparison Theorem 5 Poincare Inequality and the First Eigenvalue 6 Gradient Estimate and Harnack Inequality 7 Mean Value Inequality 8 Rei ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Contents 0 Introduction 1 First and Second Variational Formulas for Area 2 Bishop Comparison Theorem 3 BochnerWeitzenbock Formulas 4 Laplacian Comparison Theorem 5 Poincare Inequality and the First Eigenvalue 6 Gradient Estimate and Harnack Inequality 7 Mean Value Inequality 8 Reilly's Formula and Applications 9 Isoperimetric Inequalities and Sobolev Inequalities 10 Lower Bounds of Isoperimetric Inequalities 11 Harnack Inequality and Regularity Theory of De GiorgiNashMoser References 0 Introduction This set of lecture notes originated from a series of lectures given by the author at a Geometry Summer Program in 1990 at the Mathematical Sciences Research Institute in Berkeley. During the Fall quarter of 1990, the author also taught a course in G
On the Cases of Equality in Bobkov’s Inequality and Gaussian Rearrangement
"... Abstract We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the FaberKrahn inequality. ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Abstract We determine all of the cases of equality in a recent inequality of Bobkov that implies the isoperimetric inequality on Gauss space. As an application we determine all of the cases of equality in the Gauss space analog of the FaberKrahn inequality.
Mathematical analysis of the optimal habitat configurations for species persistence
, 2006
"... ..."
Differential inequalities for Riesz means and Weyltype bounds for eigenvalues
, 2007
"... We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z): = � (z − λk) σ +. ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z): = � (z − λk) σ +.
New bounds for solutions of second order elliptic partial differential equations
 Pac. J. Math
, 1958
"... 1. Introduction In a previous paper [10] the authors presented methods for determining, with arbitrary and known accuracy, the Dirichlet integral and the value at a point of a solution of Laplace's equation. These methods have the advantage that upper and lower bounds are computed simultaneously. Mo ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
1. Introduction In a previous paper [10] the authors presented methods for determining, with arbitrary and known accuracy, the Dirichlet integral and the value at a point of a solution of Laplace's equation. These methods have the advantage that upper and lower bounds are computed simultaneously. Moreover all error estimates are in terms of
Maximization of the second positive Neumann eigenvalue for planar domains
"... Abstract. We prove that the second positive Neumann eigenvalue of a bounded simplyconnected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two ide ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Abstract. We prove that the second positive Neumann eigenvalue of a bounded simplyconnected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of half this area. The estimate is sharp and attained by a sequence of domains degenerating to a union of two identical disks. In particular, this result implies the Pólya conjecture for the second Neumann eigenvalue. The proof is based on a combination of analytic and topological arguments. As a byproduct of our method we obtain an upper bound on the second eigenvalue for conformally round metrics on odddimensional spheres. 1. Introduction and